351 Midterm Exam solutions

T 2 sec b what is the fundamental frequency of the

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Unformatted text preview: n (with units)? T = 2π sec (b) What is the fundamental frequency ω of the function (with units)? f = 1/T = 1/2π ω = 2πf = 1 rad/sec (c) Is y(t) an even or odd function? odd (c) Using Fourier series definition below, find the Fourier coefficients An and Bn for y(t) where: Hint 1: note that 1 ∫ t sin(nt) = n 2 t sin( nt ) − cos(nt ) n To start, note that the average value of the y(t) over one full period is zero, and thus Ao = 0. More generally, the function is an odd function, and thus An = 0 for all n. The Bn coefficients are found as follows: € 2π −1 π Bn = ∫ (−t ) sin(nt)dt = π ∫ (t) sin(nt)dt 2π −π −π Ⱥ 2 / n n even Ⱥπ −1 Ⱥ 1 t 2 = Ⱥ 2 sin(nt ) − cos( nt ) Ⱥ = cos(nπ ) = Ⱥ Ⱥ −π n π Ⱥ n n Ⱥ −2 / n n odd € Thus the Fourier series becomes: 2 1 y ( t ) = 2 sin(t ) − sin(2 t ) + sin(3t ) − sin( 4 t ) + ... 3 2 € 3 € € 2 or : Bn = (−1) n n 30 pts 3. Consider the following Fourier series representation of a periodic function y(t): y(t) = 8 + 12sin(2π t) - 8sin(3π t) + 4sin(4π t) - sin(5π t) + … (a) Determine the fundamental frequency (in units of Hz) for y(t). Note: the true fundamental frequency should be π , since the higher-order frequencies are multiples of π , but the π term is missing from the given series, so that 2π is the first frequency term. This is an error in the exam, so either answer for ω is acceptable. Let's u se ω = 2...
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This document was uploaded on 02/06/2014.

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